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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:5. Integral Equations
Subcategory:5.3. Linear Equations of the First Kind with Constant Limits of Integration
Equation(s):$\displaystyle \int^{1}_{0}w(y)\sinh[(y-x)\sqrt{p}]dy=0 $,\hfill\break
where $p$ is a complex parameter.
Solution(s),
Transformation(s),
Integral(s)
:
Particular solution:\hfill\break
$\displaystyle w(x)=F(p)\left(C(p)e^{-x\sqrt{p}}+D(p)e^{x\sqrt{p}}+E(p)\right)$,\hfill\break
where $F(p)$ is an arbitrary function and
\begin{equation}
\left\{\begin{array}{lll}C(p)=\dfrac{4\sqrt{p}-4\sqrt{p}\,e^{\sqrt{p}}-2e^{2\sqrt{p}}+2e^{\sqrt{p}}-2e^{-\sqrt{p}}+2}{4p-e^{2\sqrt{p}}-e^{-2\sqrt{p}}+2}\\
D(p)=\dfrac{-4\sqrt{p}+4\sqrt{p}\,e^{-\sqrt{p}}-2e^{-2\sqrt{p}}-2e^{\sqrt{p}}+2e^{-\sqrt{p}}+2}{4p-e^{2\sqrt{p}}-e^{-2\sqrt{p}}+2}\\
E(p)=\dfrac{4p-4\sqrt{p}\,e^{-\sqrt{p}}+4\sqrt{p}\,e^{\sqrt{p}}+e^{2\sqrt{p}}+e^{-2\sqrt{p}}-2}{4p-e^{2\sqrt{p}}-e^{-2\sqrt{p}}+2} \end{array}\right.
\end{equation}
Remarks:Any function $w = w(y)$ that satisfies the two conditions 

$\displaystyle \int^{1}_{0}w(y)\sinh(y\sqrt{p})\,dy=0 $ \ and \
$\displaystyle \int^{1}_{0}w(y)\cosh(y\sqrt{p})\,dy=0

is a solution of the original equation.
Novelty:New equation(s) & solution(s) / integral(s)
Author/Contributor's Details
Last name:Anani
First name:Kwassi
Country:Togo
City:Lom\'e
Affiliation:Department of Mathematics, University of Lom\'e, BP 1515 Lom\'e Togo
Statistic information
Submission date:Mon 28 Dec 2009 13:18
Edits by author:0

Edit (Only for author/contributor)


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